Convergence Criteria for Operator Equations

Abstract

In this paper we deal with two nonlinear equations in real
Hilbert spaces. The rst one is of the form Au = f in which A is a
strongly monotone Lipschitz continuous operator and the second one
is of the form Au + Su = f in which S is a history-dependent operator.
The unique solvability of these equations represents well known
results. Here, our interest is in providing necessary and sucient conditions
which guarantee the convergence of an arbitrary sequence to the
solution. Our main results are gathered in Theorems 3.1, 3.2, 5.1 and 5.2.
They represents useful tools which allows us to deduce continuous dependence
results of the solution with respect to the data. They also can
be employed to prove that the solution of these equations represents the
limit of the solution of some elliptic and history-dependent variational
inequalities, respectively. We illustrate our abstract results with examples
from Solid and Contact Mechanics and provide the corresponding
mechanical interpretations.